1 Introduction

The bone remodeling cycle is a continuous process in which a portion of bone tissue is eliminated by osteoclasts through a process called bone resorption, and then osteoblasts perform bone formation consisting in the deposition of newly formed bone matrix. Osteoclasts and osteoblasts are important elements of the basic multicellular unit (BMU) which carries out the bone remodeling cycle. The BMU synchronizes in an ordered way to equilibrate bone mass levels and to maintain a functional bone structure. Such coordination is rather complex and many biochemical agents and receptors take part in this process (Florencio-Silva et al. 2015).

The RANK/RANKL/OPG pathway plays a major role in the regulation of bone remodeling. This pathway is composed of the receptor activator of nuclear factor \(\kappa \hbox {B}\) (RANKL), its receptor, RANK, and osteoprotegerin (OPG). Osteoblasts produce RANKL and OPG, while osteoclasts express RANK. Bone resorption is triggered when osteoclast precursors become active due to RANK–RANKL. To counteract this process, osteoblasts produce OPG which is a decoy receptor for RANKL that prevents RANK–RANKL bindings. The RANKL/OPG ratio determines in great way the BMU fate (Florencio-Silva et al. 2015).

Computational and mathematical models in the literature have been proposed to describe the dynamics of the bone remodeling process. The majority of the computational models are based on cellular automata algorithms to simulate bone structural adaptation (Tovar 2004; Penninger et al. 2008; Van Scoy et al. 2017). On the other hand, mathematical models are usually constructed via a system of differential equations that are based on the biochemical interactions between the BMU cells at a bone remodeling site (Komarova et al. 2003; Lemaire et al. 2004; Pivonka et al. 2008; Graham et al. 2012; Jerez and Chen 2015; Ross et al. 2017; Jerez et al. 2018). In the latter case, two families of differential models can be distinguished: one consisting of models at cellular and molecular scale following Lemaire work, and another with nonlinear models known as Komarova type equations where the nonlinearities are given by a power law (Savageau 1988) that describes the proliferation of the BMU cells.

Several factors may disrupt the cross-talk between osteoclasts and osteoblasts thereby causing bone diseases such as osteoporosis or osteopetrosis. In particular, the presence of metastatic cancer cells at the bone microenvironment is one of these factors (Florencio-Silva et al. 2015). Metastasis occurs when cancer cells spread from an initial body part, -like breast or prostate cancer-, to distant tissues, -like brain, lungs or bone-. The ‘Seed & Soil’ theory (Paget 1889) explains that cancer cells (the seeds) have preference to certain microenvironments (the soil) to metastasize. Breast and prostate cancers are the most common cancers that have high potential to form bone metastases; the former is known to cause osteolytic lesions, while the latter usually exhibits osteoblastic lesions. There is multiple evidence that supports the idea that there are many complex biochemical interactions between metastatic cells and the bone microenvironment, bringing up the hypothesis of the development of a vicious cycle that the BMU cells support cancer cells proliferation (Mundy 2002; Theriault and Theriault 2012).

Considering that one of the main causes of death in cancer patients is metastases formation (Massagué and Obenauf 2016), this disease has received much attention in the last years so to understand its mechanisms. It is known the difficulty and limitations of in vivo and in vitro bone metastasis experiments (Kwakwa et al. 2017), thus mathematical modeling may be another approach to obtain insights about the disease and try to validate some posed biological hypotheses. The number of this kind of works is considerably reduced. Lemaire extensions that model these phenomena include: Wang et al. (2011) in which multiple myeloma-induced bone disease is studied based on the bone remodeling model presented in Pivonka et al. (2008), and Farhat et al. (2017) which focuses specifically to model bone metastatic prostate cancer. Komarova type models that consider BMU-disregulation due to cancer are: Ayati et al. (2010) which focus on multiple myeloma and the BMU dynamics; Garzón-Alvaradob (2012) where both metastatic bone lesions are studied via a switch term included in the model; in Ryser et al. (2012) the OPG concentration is proposed as a key parameter mediating the bone metastasis; Coelho et al. (2016) include parathyroid hormone concentration effects and a novel way to determine the number of active osteoclasts and osteoblasts; and Jerez and Camacho (2018) where a logistic cancer equation is coupled to a Komarova bone remodeling model for describing the bone metastasis vicious cycle. From a new approach, in Dingli et al. (2009) and Warman et al. (2018) treatments for multiple myeloma and prostate cancer-induced bone disease are modeled as evolutionary games.

Despite important advances on understanding bone metastasis mechanisms, this disease is still considered incurable (Juárez et al. 2017). Palliative treatments for bone metastases are used to reduce pain and to prevent adverse consequences such as bone fractures and spinal cord compression. In this work we focus on two bone metastasis palliative treatments: denosumab and radiotherapy. Denosumab is a fully human monoclonal antibody to RANKL that acts similarly as OPG and its effectiveness in delaying the appearance of skeletal related events has been proved (Theriault and Theriault 2012). Some of the side-effects of denosumab are urinary tract infection, upper respiratory infection, hypocalcemia and osteonecrosis of the jaw (Lipton et al. 2016). On the other hand, radiotherapy is a treatment that consists in using radiation beams in localized areas of the body to kill cancer cells by damaging their DNA (Lutz et al. 2017). It is estimated that around 50% of cancer patients receive this treatment (Barker et al. 2015). Bone loss is one of the side-effects related to radiotherapy (Zhang et al. 2018). The search of optimal schedules and doses for these bone metastasis treatments continues (Chow et al. 2016; Lipton et al. 2016; Ganesh et al. 2017).

As we mentioned before, it is difficult to make in vivo or in vitro experiments to study the precise effects of bone metastasis treatments, so mathematical modeling can be an alternative approach. A way to model treatments for some disease is via an optimal control problem associated to the differential model that describes the disease. This framework is based on the idea of minimizing an expression that involves the cost of using a treatment while reducing the hazardous effects by the presence of the disease over time (Lenhart and Workman 2007). This mathematical tool was used by Lemos et al. (2016) to model treatments for multiple myeloma but has also been employed to study treatments in other biomedical models (Swan 1990; Fister et al. 1998; De Pillis and Radunskaya 2003). Here, we are interested to find optimal treatments for the denosumab and radiation therapies since significant information about effects on BMU and collateral damage is known for both treatments. The latter is essential for the mathematical modeling process and the model validation. It is important to remark that there are other novel treatments with intrinsic relevance, unfortunately we do not have enough information about them. It is our hope that this work can be used to motivate the generalization of our results for different therapies.

In the present work, we propose two optimal control models: one for denosumab treatment, and another for radiotherapy. Both of them are based on the bone metastasis Komarova type model proposed in Jerez and Camacho (2018), that describes the dynamics between cancer cells and the main BMU cells. We consider this model as a starting point since it is possible to obtain explicitly the steady-states associated with the cancer-free and cancer-invasion scenarios. Furthermore, we have conditions for the local stability of these equilibria. We also prove existence and uniqueness of solutions for both optimal control models mentioned previously. Such mathematical achievement is of paramount importance for the numerical analysis of the models. The simulations that we obtain agree qualitatively with clinical observations about the evolution of metastatic tumors on in silico experiments and on animal models. Moreover, we explore cancer-BMU dynamics for each treatment under different cancer-invasion scenarios.

The paper is organized as follows. In Sect. 2, the bone metastasis basis model is presented and discussed. Next, in Sect. 3 we propose mathematical models for two bone metastasis treatments (denosumab and radiotherapy) as optimal control problems. In Sects. 4 and 5, an optimal control framework is utilized to show existence and uniqueness of solutions, and also to pose the corresponding optimality systems. Finally, in Sect. 6 qualitative effects of treatment regimes are obtained computationally. Numerical simulations and discussions are presented.

2 Bone metastasis basis model

The model that we consider as a basis is from a previous work (Jerez and Camacho 2018). The main biological assumptions are the following:

  • There are autocrine and paracrine communications between osteoclasts (hereafter OCs) and osteoblasts (hereafter OBs) which modifies the recruitment and inhibition rates of the cells (Florencio-Silva et al. 2015). This cross-talk is approximated as a power-law (Komarova et al. 2003).

  • Bone metastatic cells (hereafter CCs) express a number of factors such as parathyroid hormone related peptide (PTHrP) and interleukins (ILs) that modify bone homeostasis (Mundy 2002; Ottewell 2016). PTHrP enhances OBs expression of RANKL, thereby increasing the number of active OCs; also, OCs resorb bone which causes the release of growth factors such as TGF-\(\beta \) than increase the production of PTHrP (Mundy 2002). Thus, we assume that the communication between OCs and CCs have positive effects on both populations (mutualism).

  • The OBs–CCs communication loop is not completely known (Ottewell 2016). In the case of an osteolytic lesion the presence of OBs may not directly impact the proliferation of CCs, and vice versa. On the other hand, for an osteoblastic lesion the OBs population may increase due to osteoblast-promoting factors produced by CCs such as IL-6. Therefore, we assume that the communication between OBs and CCs may have positive, negative or null effects on these populations.

  • Finally, we also assume that CCs population satisfies a logistic equation which is penalized by a linear elimination rate that reflects the adaptability of CCs to the bone microenvironment (Farhat et al. 2017).

For a schematic representation of the assumptions see Fig. 1.

Let us denote by C(t), B(t) and T(t) the population of OCs, OBs and CCs at a certain time t, respectively. Thus, the basis model is as follows:

$$\begin{aligned} \frac{d C(t)}{dt}&= \underbrace{\alpha _1 C(t) B(t)^{g_1}} _{\text {OCs inhibition by OBs}} - \beta _1 C(t) +\underbrace{\sigma _1 C(t) T(t),} _{\text {OCs promotion by CCs}} \end{aligned}$$
(1a)
$$\begin{aligned} \frac{d B(t)}{dt}&= \underbrace{\alpha _2 C(t)^{g_2} B(t)} _{\text {OBs promotion by OCs}} - \beta _2 B(t) + \underbrace{\sigma _2 B(t) T(t),} _{\text {CCs net effect on OBs}} \end{aligned}$$
(1b)
$$\begin{aligned} \frac{d T(t)}{dt}&= \alpha _3 T(t)\left( 1 - \frac{T(t)}{m}\right) - \beta _3 T(t) + \underbrace{\sigma _3 C(t)^{g_2} T(t)} _{\text {CCs promotion by OCs}} + \underbrace{\sigma _4 B(t)^{g_1} T(t)} _{\text {OBs net effect on CCs}}, \end{aligned}$$
(1c)

where \(\alpha _i\) are the rates of cell production for \(i=1,2,3\); \(\beta _i\) are the rates of cell removal; and \(g_1\) and \(g_2\) are the net effectiveness of paracrine factors, see Komarova et al. (2003). The exponent parameters g’s codify the effects of complex biochemical reactions on the population dynamics. We assume that \(g_1 < 0\) and \(g_2 > 0\) which correspond, respectively, to an inhibition from OBs to OCs and to a promotion of OBs due to OCs (Komarova et al. 2003; Jerez and Camacho 2018). The coefficient m is the carrying capacity of the logistic growth rate of CCs within the BMU location. The parameters \(\sigma \)’s are the proportional rates of the OCs–CCs and OBs–CCs interactions where we incorporate the possible outcomes of the vicious cycle. Depending on the signs and values of these parameters, the model exhibits osteolytic, osteoblastic or mixed lesion. It is important to remark that some of the proposed models in the literature focus only on one particular primary tumor tissue whereas we aim towards a unified model for describing and understanding more about the intertwined mechanisms of bone metastasis.

Fig. 1
figure 1

Diagrams representing simplified interaction networks in an osteolytic lesion (left) and in an osteoblastic or mixed lesion (right). Dashed lines represent inhibition/degradation (negative) effects, and solid lines represent promotion/formation (positive) effects

Full size image

2.1 Cancer-free and cancer-invasion equilibria

In Jerez and Camacho (2018), the existence of cancer-free and cancer-invasion steady-states of the model (1) is assured and explicit expressions for them are found. The cancer-free equilibrium is given by

$$\begin{aligned} \left( \ \left( \frac{\beta _2}{\alpha _2} \right) ^{\frac{1}{g_2}}, \ \left( \frac{\beta _1}{\alpha _1} \right) ^{\frac{1}{g_1}}, \ 0\ \right) , \end{aligned}$$

which is locally stable if \(g_1 g_2 < 0\) and \(\frac{\beta _2 \sigma _3}{\alpha _2}+\frac{\beta _1\sigma _4}{\alpha _1}<\beta _3- \alpha _3\). On the other hand, the cancer-invasion equilibrium, denoted by \(E_I = (C_{I}, B_{I}, T_{I})\), can be expressed as:

$$\begin{aligned} C_{I}&= \left( \frac{\alpha _1(r\beta _2+\beta _3\sigma _2-\alpha _3 \sigma _2)-\sigma _4(\beta _1\sigma _2-\beta _2\sigma _1)}{\alpha _1\alpha _2r+\alpha _1\sigma _2\sigma _3+\alpha _2 \sigma _1\sigma _4}\right) ^{\frac{1}{g_2}}, \nonumber \\ B_{I}&= \left( \frac{\alpha _2(r\beta _1+\beta _3\sigma _1 -\alpha _3\sigma _1)-\sigma _3(\beta _1\sigma _2-\beta _2\sigma _1)}{\alpha _1\alpha _2r+\alpha _1\sigma _2\sigma _3+\alpha _2\sigma _1 \sigma _4}\right) ^{\frac{1}{g_1}}, \nonumber \\ T_{I}&= \frac{\alpha _1\alpha _2\alpha _3-\alpha _1\alpha _2 \beta _3+\alpha _1\sigma _3\beta _2+\alpha _2\sigma _4\beta _1}{\alpha _1\alpha _2r+\alpha _1\sigma _2\sigma _3+\alpha _2 \sigma _1\sigma _4}, \end{aligned}$$
(2)

where \(r=\alpha _3 / m\). If \(\sigma _2<0\) and \(\sigma _4=0\), then the cancer-invasion steady-state \(E_I\) is locally stable if these three inequalities hold:

$$\begin{aligned} \frac{\beta _3-\alpha _3}{\sigma _3}<\frac{\beta _2}{\alpha _2}, \quad \frac{|\sigma _2|}{\alpha _2}<\frac{\alpha _3}{\sigma _3}, \quad \text {and} \quad C_{I} \Big (1 - \displaystyle \frac{\alpha _3}{|\sigma _2|g_1}\Big )< \frac{\beta _1}{\sigma _1}. \end{aligned}$$
(3)

Note that if we let \(\sigma _2>0\) and \(\sigma _4 =0\) then \(E_I\) is unstable.

We predict the local behavior of the solutions around the corresponding steady-states being the cancer-invasion equilibrium of great importance. If the cancer-invasion equilibrium is locally stable, then CCs may colonize the bone tissue if there are enough of these cells; if this equilibrium is unstable, then an erratic invasion (increasing oscillations) or an elimination of CCs may occur. This information is be used on Sect. 6 for the numerical simulations results.

3 Bone metastasis treatment models

In this section, we present two mathematical models for bone metastasis treatments: denosumab and radiotherapy treatments. These two treatments modify the cellular behavior of OCs, OBs and CCs. Since these three cellular populations are intertwined in an intricate communication network mediated mainly by biochemical factors, it is difficult to predict the overall dynamics of the BMU under treatments. Our goal is to understand the CCs-BMU dynamics and to address the question about the best way to dose those two treatments. For that matter, we use the optimal control framework where a cost function that includes an input (the treatment) and also undesirable variables (bone metastases) is minimized (Lenhart and Workman 2007). Such optimal control cost includes abstractly an economical cost and the side-effects of the treatment.

3.1 Denosumab treatment model

Denosumab affects the main bone remodeling signaling pathway RANK/RANKL/OPG (Florencio-Silva et al. 2015). The main biological mechanisms that we consider and their corresponding mathematical assumptions are:

  • When OBs produce RANKL and this molecule binds to RANK then OCs are activated. Denosumab neutralizes RANKL and it is used to treat bone metastasis to slow down excessive bone resorption caused by the vicious cycle CCs \(\rightarrow \) OBs \(\rightarrow \) OCs (Mundy 2002; Lipton et al. 2016). For the model, we propose that denosumab alters the cell activity of OCs in a proportional way.

  • Denosumab has various side-effects on the patient, such as osteonecrosis of the jaw (Theriault and Theriault 2012). We propose to combine these side-effects and the economical cost of this treatment in a function that measures both components.

Let us denote by \(u_D(t)\) the effect of denosumab on the activity of OCs. Considering the previous assumptions, we propose the following denosumab model:

$$\begin{aligned} \min _{u_D} J\left( u_D(t) \right) \quad \text {where} \quad J\left( u_D(t) \right) = \int _{0}^{t_f}\left( \smash {\underbrace{w_D u_D(t)^2} _{\text {DT cost and side-effects}}} + T(t)^2 \right) dt \end{aligned}$$
(4a)

subject to:

$$\begin{aligned}&\frac{d C(t)}{dt} = \alpha _1 \underbrace{\left( 1 - u_D(t)\right) }_ {\text {DT effects on OCs}} C(t) B(t)^{g_1} - \beta _1 C(t) + \sigma _1 C(t) T(t), \end{aligned}$$
(4b)
$$\begin{aligned}&\frac{d B(t)}{dt} = \alpha _2 C(t)^{g_2} B(t) - \beta _2 B(t) + \sigma _2 B(t) T(t), \end{aligned}$$
(4c)
$$\begin{aligned}&\frac{d T(t)}{dt} = \alpha _3 T(t)\left( 1 - \frac{T(t)}{m}\right) - \beta _3 T(t) + \sigma _3 C(t)^{g_2} T(t) + \sigma _4 B(t)^{g_1} T(t), \end{aligned}$$
(4d)
$$\begin{aligned}&C(0),\ B(0),\ T(0) > 0 \quad \text {given,} \end{aligned}$$
(4e)
$$\begin{aligned}&0 \le u_D(t) \le u_D^{\max } < 1 \quad \text {for all} \quad 0 \le t \le t_f, \end{aligned}$$
(4f)

where DT stands for denosumab therapy, \(u_D \equiv 0\) means that no denosumab is applied and \(u_D \equiv u_D^{\max }\) reflects the maximum effectivity of denosumab in nullifying the activity of OCs. The cost functional \(J\left( u_D(t) \right) \) measures the economical cost and side-effects of using denosumab and stores the net side-effects due to the presence of CCs. To construct the cost functional, we follow the standard notion of non-linear cost functionals (Lenhart and Workman 2007). In particular, the cost per unit of time of the presence of CCs is measured by \(T(t)^2\), while the use of denosumab produce a cost per unit of time in term of its effectivity \(u_D(t)^2\). In the cost functional (4a), the parameter \(w_D\) is a weight control parameter that represents the normalized cost of using denosumab. We assume a fixed time window from a starting day 0 to a final day denoted by \(t_f\).

3.2 Radiotherapy treatment model

Another treatment option for patients with bone metastasis is radiotherapy. In Zhang et al. (2018) the authors offers a landscape of what is known about the effects of radiation on the bone cells, particularly on OCs and OBs. We now mention the key biological aspects of radiotherapy and the associated mathematical assumptions:

  • The main action of radiotherapy is to disrupt CCs proliferation by damaging their DNA; however, it also affects non-cancerous cells of the body like OCs and OBs (Vakaet and Boterberg 2004; Brenner 2008; Zhang et al. 2018). Therefore, we propose that radiation increases the elimination rates of OCs, OBs and CCs.

  • Radiation may cause haematological toxicity, nausea and vomiting (Chow et al. 2016). Taking into account this fact, we include the use of radiation with a function that measures these effects along with the economical cost of the treatment.

  • It has been observed that irradiation may impairs bone remodeling in the long run (Oest et al. 2015; Zhang et al. 2018). We propose that the effects of a radiation dose decay exponentially rather than instantly on the BMU.

Let us denote by \(u_R(t)\) the cell-killing rate due to radiation on the BMU. Considering the mentioned assumptions, we propose the following radiotherapy model:

$$\begin{aligned} \min _{u_R} J\left( u_R(t) \right) \quad \text {where} \quad J\left( u_R(t) \right) = \int _{0}^{t_f}\left( \smash {\underbrace{w_R u_R(t)^2} _{\text {RT cost and side-effects}}} + T(t)^2 \right) dt, \end{aligned}$$
(5a)

subject to:

$$\begin{aligned}&\frac{d C(t)}{dt} = \alpha _1 C(t) B(t)^{g_1} - \underbrace{\left( \beta _1 + \psi _1 u_R(t)\right) }_ {\text {RT effects on OCs}} C(t) + \sigma _1 C(t) T(t), \end{aligned}$$
(5b)
$$\begin{aligned}&\frac{d B(t)}{dt} = \alpha _2 C(t)^{g_2} B(t) - \underbrace{\left( \beta _2 + \psi _2 u_R(t)\right) }_ {\text {RT effects on OBs}} B(t) + \sigma _2 B(t) T(t), \end{aligned}$$
(5c)
$$\begin{aligned}&\frac{d T(t)}{dt} = \alpha _3 T(t)\left( 1 - \frac{T(t)}{m}\right) - \underbrace{\left( \beta _3+u_R(t)\right) }_ {\text {RT effects on CCs}} T(t) + \sigma _3 C(t)^{g_2} T(t) + \sigma _4 B(t)^{g_1} T(t), \end{aligned}$$
(5d)
$$\begin{aligned}&C(0),\ B(0),\ T(0) > 0 \quad \text {given,} \end{aligned}$$
(5e)
$$\begin{aligned}&0 \le u_R(t) \le u_R^{\max } \quad \text {for all} \quad 0 \le t \le t_f. \end{aligned}$$
(5f)

where RT stands for radiotherapy, the parameters \(u_R^{\max }\) and \(w_R\) are the analogues of \(u_D^{\max }\) and \(w_D\) as in the denosumab case. Here, we propose to increase linearly the CCs elimination rate \(\beta _3\) by \(u_R(t)\). To account for adverse effects on the proliferation of OCs and OBs, we assume that their respective elimination rates are also affected by radiation as well. For that reason, we introduce the coefficients \(\psi _1\) and \(\psi _2\) that act as rescaling parameters of the radiation effect \(u_R(t)\) on OCs and OBs, respectively.

4 Optimal solution for the denosumab model

In order to explore these bone metastasis treatment models, it is important to guarantee the existence of optimal solutions, \(u_D(t)\) and \(u_R(t)\), that satisfy the corresponding problems (4) and (5). In this section, we prove the existence and also the uniqueness of an optimal control solution to the denosumab treatment model (4), and we discuss how the radiotherapy model (5) has analogous results.

4.1 Existence of optimal solutions

We are interested in studying the effects of bone metastasis treatments on the BMU when the tumor has the potential to establish or has already established a successful invasion. To accomplish this, we assume two possible scenarios related to the boundness and positivity of the solutions for the non-treatment model (1):

Assumption A

The cancer-invasion equilibrium is locally stable and the initial conditions are located nearby.

Assumption B

The cancer-invasion equilibrium is locally unstable, then we assume a priori that the state variables remain inside a compact subset.

Assumption A is valid if the steady-state (2) satisfies conditions (3). Assumption B is a reasonable assumption that has been proposed in other optimal control problems, see for instance (Bara et al. 2017). In either case, these assumptions lead us to consider a compact subset near the cancer-invasion equilibrium from which the solutions stay in that subset for every positive time \(t > 0\).

Now, let \(\varOmega \) be a compact subset of the natural domain of model (1) defined by \(\varOmega ^+ = \{(C(t),B(t),T(t))\ |\ x_i > 0,\ i=1,2,3 \}\). In this subset, the state variables (C(t), B(t), T(t)) are uniformly bounded because \(\varOmega \) is a compact subset of \(\varOmega ^ +\). This means that, for all \(t\in [0,t_f]\), we have

$$\begin{aligned} C(t) \le C^{\max } \quad \text {and} \quad B(t) \ge B^{\min }. \end{aligned}$$

Let us denote the state variables as \(x(t) = \left( C(t), B(t), T(t) \right) \). To prove the existence of an optimal control we employ Theorem 4.1 from Fleming and Rishel (1975). Such result states that the the following conditions are sufficient to guarantee the existence of an optimal control solution for (4):

(H1):

The right-hand side of the model (4b)–(4d) is composed of continuous functions, and for each one of these functions \(f_i\) there exist positive constants \(C_1, C_2\) such that \(|f_i(t,x,u_D)| \le C_1 (1 + |x| + |u_D|)\) and \(|f_i(t,\bar{x},u_D) - f_i(t,x,u_D)| \le C_2 |\bar{x} - x| (1 + |u_D|)\) for all \(0 \le t \le T\) and \(i=1,2,3\).

(H2):

There exists at least one pair \(\left( x(t),u_D(t) \right) \) satisfying both (4b)–(4d) and (4f).

(H3):

The set of admissible controls is closed and convex.

(H4):

The right-hand side of the model is bounded above by a sum of the states and the control, and it can be written as a linear function with respect the control.

(H5):

The integrand of cost functional is convex with respect the control and it is bounded above by \(C_3|u_D|^{n} - C_4\) for some fixed \(C_3 > 0\), \(C_4 \in \mathbb {R}\) and \(n > 1\).

We proceed to show such conditions.

Lemma 1

The model (4) satisfies (H1).

Proof

It is straightforward since the model functions are of class \(C^2\) in \(\varOmega \). \(\square \)

Lemma 2

There exists at least one pair \((x(t),u_D(t))\) with \(u_D\in \mathcal {U}\) such that Eq. (4b)–(4d) is satisfied.

Proof

The condition (H2) is satisfied by the Carathéodory’s existence theorem (see Theorem 9.2.1 from Lukes 1982), which guarantees the existence of solutions for Cauchy problems. \(\square \)

Lemma 3

The set of admissible controls is closed and convex.

Proof

Since \(0 \le u_D(t) \le u_D^{\max }\) then the lemma requirements are satisfied. \(\square \)

Lemma 4

The right-hand side of (4b)–(4d) is continuous, also it is bounded from above by a sum of the states and the control, and it can be written as a linear function of the control.

Proof

Let \(f(t,x,u_D)\) be the vector function defined by the right-hand side of (4b)–(4d). As mentioned above, f is continuous on \(\varOmega \). Now we have to find suitable bounds for the states. Since \(0 < C(t) \le C^{\max }\), \(B^{\min } \le B(t)\) and \(0 < T(t) \le C^{\max }\), where the constants \(C^{\max }\), \(B^{\min }\) and \(C^{\max }\) come from the definition of the domain \(\varOmega \), then:

$$\begin{aligned} \frac{d C(t)}{dt}&= \alpha _1 C(t) B(t)^{g_1} (1-u_D) - \beta _1 C(t) + \sigma _1 C(t) T(t) \nonumber \\&\le \alpha _1 C(t) B(t)^{g_1} - \beta _1 C(t) + \sigma _1 C(t) T(t) \nonumber \\&\le \alpha _1 C(t) m_1 - \beta _1 C(t) + \sigma _1 C(t) T(t) \quad (g_1 < 0 \text { and } m_1 := (B^{\min })^{g_1}) \nonumber \\&\le \alpha _1 C(t) m_1 + \sigma _1 C(t) T(t) \nonumber \\&= C(t) (\alpha _1 m_1 + \sigma _1 C^{\max }), \end{aligned}$$
(6)

so dC(t) / dt is bounded from above by the linear Eq. (6). Similarly,

$$\begin{aligned} \frac{d B(t)}{dt} \le B(t) (\alpha _2 m_2 + \sigma _2 C^{\max }), \end{aligned}$$

where \(m_2 := (C^{\max })^{g_2}\). Taking into account that: \(\sigma _2 \ge 0\) or \(\sigma _2 < 0\), then

$$\begin{aligned} \frac{d B(t)}{dt} \le \left\{ \begin{array}{ll} B(t) (\alpha _2 m_2 + \sigma _2 C^{\max }) &{}\quad \text {if } \sigma _2 \ge 0 \\ B(t) \alpha _2 m_2 &{}\quad \text {if } \sigma _2 < 0. \end{array}\right. \end{aligned}$$
(7)

Hence, B(t) is also bounded from above by a linear equation. Analogous to C(t) and B(t) and considering that \(\sigma _4 \le 0\), we have

$$\begin{aligned} \frac{d T(t)}{dt} \le T(t) \left( \alpha _3 - \beta _3 + \sigma _3 m_2\right) . \end{aligned}$$
(8)

From inequalities (6)–(8), we know that the model is bounded from above by a linear system. Thus, the solutions are bounded for a finite final time. Observe that there are two cases given by the sign of \(\sigma _2\). These inequalities, together with the triangle inequality, also give:

$$\begin{aligned}&|f(t,x,u_D)| \nonumber \\&\quad = \left| \left( \frac{d C}{dt}, \frac{d B}{dt}, \frac{d T}{dt} \right) ^{\intercal } \right| \nonumber \\&\quad = \left| \begin{pmatrix} \alpha _1 m_1 + \sigma _1 C^{\max } &{} 0 &{} 0 \\ 0 &{} \alpha _2 m_2 + \sigma _2 C^{\max } &{} 0 \\ 0 &{} 0 &{} \alpha _3 - \beta _3 + \sigma _3 m_2 \end{pmatrix} \begin{pmatrix} C \\ B \\ T \end{pmatrix}\right| + \left| \begin{pmatrix} \alpha _1 C^{\max } m_2 \\ 0 \\ 0 \end{pmatrix} u_D \right| \nonumber \\&\quad \le C_1(|x| + |u_D|), \end{aligned}$$
(9)

where \(C_1\) depends on the model parameters and the bounds of its solutions. The case for \(\sigma _2 < 0\) is analogous. \(\square \)

Lemma 5

The integrand from (4a) is convex in the control, and it is bounded from above by \(C_3 |u_D|^{n} - C_4\) with \(C_3 > 0\) and \(n > 1\).

Proof

The integrand \(L(t,x,u_D) = T(t)^2 + w_D u_D^2\) is convex respect \(u_D\), and also

$$\begin{aligned} L(t,x,u_D) = T(t)^2 + w_D u_D^2 \ge w_D u_D^2 = C_3 |u_D|^{n}, \end{aligned}$$
(10)

with \(C_3 = w_D > 0\) and \(n = 2 > 1\). \(\square \)

The above discussion allows us to state the following result:

Theorem 1

The denosumab treatment model (4), considering the domain \(\varOmega \), has an optimal control \(u_D^*\). \(\square \)

4.2 Optimality system

Under Assumptions A or B we have prove the existence of at least one optimal control. Here, we use Pontryagin’s Maximum Principle (Pontryagin et al. 1962; Lenhart and Workman 2007) to derive necessary conditions that every optimal control satisfies. Let H be the Hamiltonian defined by

$$\begin{aligned} H&= \left( \alpha _1 C(t) B(t)^{g_1} \left( 1 - u_D(t) \right) - \beta _1 C(t) + \sigma _1 C(t) T(t) \right) \lambda _1(t) \nonumber \\&\quad + \left( \alpha _2 C(t)^{g_2} B(t) - \beta _2 B(t) + \sigma _2 B(t) T(t) \right) \lambda _2(t) \nonumber \\&\quad + \left( \alpha _3 T(t) \left( 1 - \frac{T(t)}{m}\right) - \beta _3 T(t) + \sigma _3 C(t)^{g_2} T(t) \right) \lambda _3(t) \nonumber \\&\quad + T(t)^2 + w_D u_D(t)^2. \end{aligned}$$
(11)

From (11) we get the following adjoint system for denosumab model (4):

$$\begin{aligned} \frac{d \lambda _1}{dt}&= - \lambda _1 \left( \alpha _1 B^{g_1} \left( 1 - u_D \right) - \beta _1 + \sigma _1 T \right) - \lambda _2 \left( \alpha _2 g_2 C^{g_2-1} B \right) \nonumber \\&\quad - \lambda _3 \left( \sigma _3 g_2 C^{g_2-1} T\right) , \end{aligned}$$
(12a)
$$\begin{aligned} \frac{d \lambda _2}{dt}&= - \lambda _1 \left( \alpha _1 g_1 C B^{g_1-1} \left( 1 - u_D \right) \right) - \lambda _2 \left( \alpha _2 C^{g_2} - \beta _2 + \sigma _2 T \right) , \end{aligned}$$
(12b)
$$\begin{aligned} \frac{d \lambda _3}{dt}&= - \lambda _1 \sigma _1 C - \lambda _2 \sigma _2 B - \lambda _3 \left( \alpha _3 \left( 1 - \frac{2 T}{m} \right) - \beta _3 + \sigma _3 C^{g_2} \right) - 2 T, \end{aligned}$$
(12c)
$$\begin{aligned} \lambda _1(t_f)&= \lambda _2(t_f) = \lambda _3(t_f) = 0. \end{aligned}$$
(12d)

The optimality condition for (4), obtained also by means of the Pontryagin’s Maximum Principle, is the following:

$$\begin{aligned} u_D^*(t) = \max \left\{ 0, \min \left\{ 1, \frac{\alpha _1 C(t) B(t)^{g_1}\lambda _1(t)}{2 w_D}\right\} \right\} , \end{aligned}$$
(13)

which is the characterization of every optimal solution \(u_D^*\) for (4) in terms of the state variables, the adjoint variables and the parameters of the model. A direct use of the Maximum Principle gives us:

Theorem 2

Let \(u_D^*\) be an optimal control for the denosumab model (4) and \(x=(C,B,T)\) its associated state-variable. Then there exist functions \(\lambda _1(t)\), \(\lambda _2(t)\) and \(\lambda _3(t)\) that satisfy the adjoint system (12). Also, the optimal control \(u_D^*\) satisfies the optimality condition (13). \(\square \)

4.3 Uniqueness of optimal solutions

In order to prove the uniqueness of an optimal control problem, we follow the steps proposed in Fister et al. (1998) and state the next theorem:

Theorem 3

There exists a final time \(t_f\) such that the model (4) has a unique optimal control solution.

Proof

See Appendix. \(\square \)

5 Optimal solution for the radiotherapy model

Besides the denosumab treatment, we also explore the radiotherapy effects on the dynamics of the bone metastasis model. In this section, we give the optimality system and discuss about the existence and uniqueness of the optimal control for the radiotherapy model (5).

5.1 Existence of optimal solutions

The five lemmas for the existence of solutions of the denosumab model are proved in the same way for the radiotherapy treatment model, since they have a similar structure: an a priori bounded domain, a bounded control \(u_R\), and the model is linear respect to \(u_R\). Thus, similar algebraic manipulations give the existence of solutions for the radiotherapy model.

5.2 Optimality system

Analogously as in the denosumab model, using the Maximum Principle we obtain the following optimality system for the radiotherapy model (5):

$$\begin{aligned} \frac{d C}{dt}&= \alpha _1 C B^{g_1} - (\beta _1 + \psi _1 u_R^*) C + \sigma _1 C T, \end{aligned}$$
(14a)
$$\begin{aligned} \frac{d B}{dt}&= \alpha _2 C^{g_2} B - (\beta _2 + \psi _2 u_R^*) B + \sigma _2 B T, \end{aligned}$$
(14b)
$$\begin{aligned} \frac{d T}{dt}&= \alpha _3 T\left( 1 - \frac{T}{m}\right) - (\beta _3+u_R^*) T + \sigma _3 C^{g_2} T + \sigma _4 B^{g_1} T, \end{aligned}$$
(14c)
$$\begin{aligned} \frac{d \lambda _1}{dt}&= -\alpha _{2} g_{2} \lambda _{2} C^{g_{2} - 1} B - \sigma _{3} g_{2} \lambda _{3} C^{g_{2} - 1} T - \left( \alpha _{1} B^{g_{1}} + \sigma _{1} T - \beta _{1} - \psi _1 u_R^* \right) \lambda _{1}, \end{aligned}$$
(14d)
$$\begin{aligned} \frac{d \lambda _2}{dt}&= -\alpha _{1} g_{1} \lambda _{1} C B^{g_{1} - 1} - \sigma _{4} g_{1} \lambda _{3} B^{g_{1} - 1} T - {\left( \alpha _{2} C^{g_{2}} + \sigma _{2} T - \beta _{2} - \psi _2 u_R ^*\right) } \lambda _{2}, \end{aligned}$$
(14e)
$$\begin{aligned} \frac{d \lambda _3}{dt}&= -\sigma _{1} \lambda _{1} C - \sigma _{2} \lambda _{2} B - {\left( \sigma _{3} C(t)^{g_{2}} + \sigma _{4} B^{g_{1}} - \alpha _{3} {\left( \frac{T}{m} - 1\right) } \right. } \nonumber \\&\quad {\left. - \beta _{3} - u_R^* - \frac{\alpha _{3} T}{m}\right) } \lambda _{3} - 2 T,\nonumber \\&\quad C(0), B(0), T(0) \quad \text {given}, \end{aligned}$$
(14f)
$$\begin{aligned}&\quad \lambda _1(t_f) = \lambda _2(t_f) = \lambda _3(t_f) = 0, \end{aligned}$$
(14g)
$$\begin{aligned} u_R^*(t)&= \max \left\{ 0, \min \left\{ 1, \frac{\psi _1 C(t) \lambda _1(t) + \psi _2 B(t) \lambda _2(t) + T(t) \lambda _3(t)}{2 w_R} \right\} \right\} . \end{aligned}$$
(14h)

5.3 Uniqueness of optimal solutions

Uniqueness of optimal solutions for the radiotherapy model is analogous to the denosumab case since changes in the model (ODEs, cost functional or control restrictions) produce a similar effect in the optimality system (adjoint system and optimality condition).

6 Numerical results and discussion

The existence and uniqueness of solutions guarantee that a convergent numerical method will get the approximation to the unique optimal solution. Taking advantage of that, here we use the forward-backward sweep method (FBSM) (Lenhart and Workman 2007). This numerical scheme is based upon the iterative use of the Maximum Principle. Considering the optimal control problem for the denosumab model (4) together with its adjoint system (12) and the optimality condition (13), the steps involved in the FBSM are the following:

  1. 1.

    Propose an initial control \(u_D^0(t)\).

  2. 2.

    Solve forward in time the state variable system (4).

  3. 3.

    Solve backward in time the adjoint system (12).

  4. 4.

    Use the Maximum Principle to get a new control update \(u_D^k\) for step k. Here, we consider (Lenhart and Workman 2007):

    $$\begin{aligned} u_D^k(t) \hookleftarrow \mu u_D^k(t) + (1-\mu ) u_D^{k-1}(t). \end{aligned}$$

    If a convergence criteria is met, e.g., the control update is close to previous control, then STOP; else, return to STEP 2.

For solving the forward and backward ODEs, we used a fourth order Runge–Kutta scheme with variable time step. Let us point out that the problem about the convergence of the FBSM is discussed in McAsey et al. (2012). In that work, the authors prove results about the convergence of the FBSM, and the main hypotheses required to guarantee such convergence are Lipschitz conditions, an appropriate length for the integration interval and boundedness of the adjoint system. Some of these conditions, in particular the last one, are difficult to satisfy a priori because of the non-linearities of the treatment models.

To achieve convergence, here we set the maximum value of the control \(u_D\) to \(u_D^{\max } = 0.6\) instead of using \(u_D^{\max } = 1\). From a modeling perspective, this means that we assume that the treatment does not have a complete effectiveness. An example of this approach, used in a different problem, can be found in Stephenson et al. (2017). Also, we initialized the control \(u^0\) for the FBSM as \(u^0 \equiv u_D^{\max }\). We have convergence of the simulations when \(\mu \) takes values within the interval \(0.15 \le \mu \le 0.6\).

Table 1 Fixed global parameter values, see Sect. 6.1 for discussion
Full size table

6.1 Parameters for numerical simulations

After exploring numerous combinations for the parameter values, and in an effort to agree with experimental data (Komarova et al. 2003; Farhat et al. 2017; Araujo et al. 2014), we considered a fixed set of values for certain parameters depicted in Table 1 for models (4) and (5). Next, we discuss their selection:

  • Initial condition (C(0), B(0), T(0)): We chose the initial conditions \(C(0) = 4 \times 10^{-6}\), \(B(0) = 4\) and \(T(0) = 1000\) according with (Komarova et al. 2003; Jerez and Chen 2015).

  • The net effectiveness parameters \(g_1\) and \(g_2\): We preserved OBs-induced inhibition on OCs by taking \(g_1 = -0.3\) as in Jerez and Chen (2015). However, in this work we increased \(g_2\) from 0.5 (Jerez and Camacho 2018) to 0.7 in order to have a more active remodeling process, that is, larger amplitudes and shorter periods for the OCs and OBs solutions without cancer.

  • The cell activity parameters \(\alpha _1\) and \(\alpha _2\): By fixing values of \(g_1\) and \(g_2\) we estimated through trial and error these parameters to obtain standard numbers of OCs and OBs with and without cancer.

  • The elimination rates \(\beta _1\) and \(\beta _2\): They are proposed as in Komarova et al. (2003).

  • Coefficients for CCs: The elimination rate \(\beta _3\) is considered zero since we include its effect in \(\alpha _3\), having a net production rate for CCs. For the production rate of CCs, \(\alpha _3\), we take a realistic interval based on (Ayati et al. 2010; Farhat et al. 2017). We estimated a normal carrying capacity for the CCs, m, using (Farhat et al. 2017).

  • Rates of the OCs-CCs and OBs–CCs interactions \(\sigma _i\)\((i=1,\ldots ,4)\): They are difficult to estimate a priori since we do not have experimental data. So, a trial and error parameter space exploration was made. Election criteria for their values took into consideration the cancer-invasion equilibrium value (2) together with its stability conditions (3), see Fig. 2. Values for these parameters were discarded when erratic numbers of OCs or OBs were presented.

  • The radiotherapy control parameters \(\psi _j\)\((j=1,2)\): They are equal to 1 as a first approach. This selection arises from the observation that other values do not change the qualitative behavior of the optimal solutions in our parameter space exploration.

To complement these fixed parameters, we varied the values for \(\alpha _3\), \(\sigma _1\), \(\sigma _3\) and \(\sigma _4\) and obtained three metastatic invasion scenarios; their values are condensed in Table 2. These three scenarios are of biological relevance because they are associated with different dynamics of invasion which may be presented in different physiological settings (Mundy 2002; Ottewell 2016).

Fig. 2
figure 2

Bifurcation diagrams corresponding to Scenarios 1–3. The y-axis is the T(t)-coordinate of the corresponding cancer-invasion steady-state \(T_I\) given in (2)–(3). Left. Denosumab treatment model with bifurcation parameter \(u_D\). Right. Radiotherapy model with bifurcation parameter \(u_R\)

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Table 2 Parameters for three different scenarios of the metastatic invasion
Full size table

6.2 Denosumab treatment

As mentioned before, the function of denosumab is to inhibit osteoclasts activation through RANK-RANKL bindings. Since it is only a palliative treatment, it is expected that only osteoclasts-dependent bone metastatic invasions are heavily affected by means of denosumab administration.

6.3 Scenario 1: Aggressive metastasis

In this case, the proliferation of CCs is rapid and so this scenario represents an aggressive type of bone metastasis tumor. It may be noted that we are considering in this case that OBs do not affect directly the CCs dynamics (\(\sigma _4 = 0\)). In Fig. 2, we show that the cancer-invasion steady-state is unstable for all relevant values of \(\alpha _1\) (from 0 to its actual value 0.5). Also, it can be noted that the cancer-coordinate does not change its value in this range. That means that the steady-state is not changed in this coordinate albeit a treatment is applied.

Fig. 3
figure 3

Scenario 1 for denosumab treatment model for three different cost weight values

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In Fig. 3, it can be noted that the optimal treatment obtained is an aggressive one: the maximum amount of effectivity (\(u_{\max }\)) is maintained during almost all the time range, and then is suspended abruptly. Also, we observe that OCs wave is displaced in time but maintains its normal amplitude. Due to the cross-regulation between bone cell populations, this will cause also a displacement of the OBs wave (not seen in the selected time range). However, CCs are indifferent to the decrease of OCs and OBs. We assume that the proliferation \(\alpha _3\) is high enough to pull the dynamics of CCs away from the bone resorption contribution \(\sigma _1\). Thus, this scenario presents a metastatic tumor that does not rely completely on the BMU dynamics. As such, the treatment shows to be ineffective, which is clinically observed on advanced aggressive bone metastatic patients (Coelho et al. 2016).

6.4 Scenario 2: Osteoclasts-dependent metastasis

In this case the metastatic tumor has a noticeable change when the OCs waves are reduced. In Fig. 2 there is a similar bifurcation diagram as in scenario 1: stability and value of the cancer coordinate do not change with variations of the \(\alpha _1\) parameter.

Fig. 4
figure 4

Scenario 2 for denosumab treatment model

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It can be noted in Fig. 4, that the optimal solution obtained has a maximum amplitude (having a value around 0.4) almost the half of the value of \(u_{\max }\); also, its shape is similar to the one of the OCs wave. The treatment applied causes the OCs wave to diminish in amplitude but preserves its appearance in time. The effect of this is also a smaller amplitude on the OBs wave. By contrast from the first case, the CCs population has a visible effect (reduction of 5% compared with the non-treatment case). This is a metastatic invasion that depends more on the BMU dynamics than the previous scenario. The parameters election for this scenario suggests that the tumor depends on the OCs dynamics rather than on the OBs evolution (\(\sigma _3 > 0\) and \(\sigma _4 = 0\)). In this case, treatment regimes show to be more effective than the previous scenario; this is due to the OCs-activity dependence of the CCs proliferation.

6.5 Scenario 3: Slow, BMU-dependent metastasis

In Fig. 2 a different bifurcation diagram is present for scenario 3. The steady-state branch now goes to zero when the parameter \(\alpha _1\) decreases, and the stability does not change from \(\alpha _1 = 0.5\) to \(\alpha _1 = 0.2\). This suggests that a strong inhibition over the OCs activation would lead the CCs population to be eradicated.

In Fig. 5, however, the optimal solution obtained is to apply the treatment almost to the lowest levels. The amplitudes of the OCs and OBs waves decrease by a small amount and the CCs burden maintains nearly to the same quantity. This simulation suggests that increasing the use of the treatment, even if it drops CCs numbers to a lower level, is more expensive than the cost produced for the reduction of the CCs population with stronger doses. In other words, the metastatic tumor grows or reduces slowly. It is predicted that if the value of \(w_D\) is lowered then the optimal solution would take into account a stronger inhibition on OCs.

Fig. 5
figure 5

Scenario 3 for denosumab treatment model

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Fig. 6
figure 6

Scenario 3 for denosumab treatment. The cost parameter is changed to \(w_D=1 \times 10^{4}\)

Full size image

We tested such hypothesis by changing the value of \(w_D\) to \(1\times 10^{4}\). The results from the simulation are shown in Fig. 6. The inhibition of OCs activation lasts longer and the metastatic burden decreases more than with the previous values of \(w_D\). Yet, the OCs wave appearance delays considerable due to the inhibition applied by the treatment and so the amplitude of the OCs wave increases considerably (3-fold the usual). Thus, the OBs wave also increases to drastic levels compared to the untreated case. Both abnormal waves have a positive effect on the metastatic tumor at the end of the treatment. Here, we note that it may be interesting to study how the secondary effects of a treatment may impact the appearance and the amplitude of the OCs and OBs waves.

6.6 Radiotherapy treatment

We adapted the FBSM to find approximations of optimal solutions also for the radiotherapy model (14a). For this model convergence of the method was less sensitive than the denosumab treatment model. In Figs. 7, 8 and 9 are shown the same three scenarios given by Tables 1 and 2 but using the radiotherapy treatment model (14a).

6.7 Scenario 1: Aggressive metastasis

In the first scenario, in Fig. 7), there are two optimal solutions that are aggressive (maximum radiation effectivity present). The effects on the OCs and OBs populations are similar under the three treatment regimes. The CCs population decreases its proliferation rate considerably but maintains a prevalent level. It can be noticed that during the OCs wave activation there is a slight increase in the CCs population.

6.8 Scenario 2: Osteoclasts-dependent metastasis

In the second scenario, Fig. 8, only one treatment regimen reaches the maximum value 0.015. The difference on the effects of applying the treatment are more noticeable than in the first scenario. But, as in the previous one, OCs and OBs populations do not change much. It is interesting to observe that under the first two treatments (with \(w_R=1 \times 10^{9}\) and \(w_R=1 \times 10^{10}\)) the CCs population shows a slight increase during the remodeling wave activation; in these two regimes the CCs population drops down to lower levels.

Fig. 7
figure 7

Scenario 1 for radiotherapy treatment model for three different cost weight values

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Fig. 8
figure 8

Scenario 2 for radiotherapy treatment model

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Fig. 9
figure 9

Scenario 3 for radiotherapy treatment model

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6.9 Scenario 3: Slow, BMU-dependent metastasis

The third scenario, Fig. 9, is rather similar to the second one. The main difference is that, in this case, the CCs population does not show a significant increase during activation of OCs and OBs waves.

6.10 Summary of numerical simulations

In this section, we explored three scenarios corresponding to different responses of the bone metastatic tumor to the microenvironment: a reckless tumor, a tumor that relies on bone resorption, and a tumor that depends on both bone cell populations.

By bifurcation diagram from Fig. 2, we could predict that, for the particular values of the model parameters, the denosumab treatment is only effective in the third scenario, which is the BMU dependent tumor. And even so, when the weight parameter \(w_D\) is large enough then a stronger denosumab treatment is more expensive even if it has the ability to reduce the tumor size. The simulations from Figs. 3, 4, 5 and 6 reflect some possible outputs for the cellular dynamics changes under a denosumab treatment that seeks to minimize economical cost and side-effects.

Similarly, the radiotherapy model was numerically explored in Figs. 7, 8, and 9. In contrast to the denosumab treatment model, the bifurcation diagram reveals that radiation has a high potential to bring down the tumor at equilibrium. Another difference in comparison to the denosumab treatment model is that for the radiotherapy model we assumed that radiation also alters the elimination rates of the bone cells. The numerical simulations show interesting cellular dynamics.

Even though that in appearance the radiotherapy shows a better performance in reducing the bone metastasis tumor that the denosumab treatment, it is important to point out that in the numerical simulations radiation has weight parameters that surpass by at least three orders of magnitude the ones assigned for the denosumab treatment. This implies that radiation was assigned a higher cost in terms of economical cost and also from the point of view of the side-effects. Increasing the weight parameters for radiation would translate in affecting less the growth of the tumor.

7 Conclusions

Bone metastasis disease is considered incurable. At the time, several palliative treatments aim towards slowing down the growth of secondary tumors in the bone. These treatments may increase their performance if optimal dosages could be found.

In this work, two bone metastasis treatment models are presented extending a previous model that studies the dynamics of the BMU cells (osteoclasts and osteoblasts) and bone metastatic cancer cells. Having explicit expressions for the steady-states and conditions for their local stability, an exploration of parameters was made in order to find a multitude of dynamics that have already been seen qualitatively in experiments. The objective was to find and analyze treatment regimes for two kind of treatments (denosumab and radiotherapy) and also to study their effects on the cellular dynamics. An optimal control approach incorporates a cost function of the treatment use reflecting the economical cost and also side-effects.

Having knowledge of how the cancer-invasion equilibrium changes locally according to variations of certain parameters is very useful. We presented simulations that allowed us to verify if denosumab and radiation treatments are effective to reduce cancer cell levels. We considered a number of possible relevant scenarios of bone metastatic evolution under treatment. In all of them, the optimal treatment regime obtained depends on the manipulation of the remodeling wave (amplitude or time of appearance). In some cases, the cancer cell populations are already aggressive enough to be influenced by the inhibition of osteoclasts; in other cases, their dependence is rather low and so the metastatic burden is not decreased enough, or it may be decreased at cost of disrupting the osteoclasts-osteoblasts cross-talk in a negative way. Thus, the inhibition of osteoclasts is not always the best answer depending on the type of metastatic cancer residing in the bone microenvironment. As predicted, denosumab treatment poses a weak choice in terms of controlling the growth of the tumor in general. Radiation treatment has a higher potential than the previous one, but the effects on the bone cells still need to be analyzed. Also, radiation has long-run side-effects and important economical cost that limit its applicability. A treatment that respects the natural microenvironment while attacking cancer cells would be more effective and desirable.

We have also observed that the weight parameter of the cost functional does have an important role: if the cost of use of the treatment is rather high then the optimal solution would be to apply low-dose treatments but avoiding a reduction of the metastatic burden.

Thus, it would be interesting to explore alternative treatment strategies by considering other potential control terms. For example, it could be interesting to explore a mathematical model that uses both controls, denosumab and radiation, and evaluate if they can act in a more effective way towards eradicating bone metastasis. Another alternative would be to explore a control that halts interactions between cancer and osteoclasts and osteoblasts by inhibiting other key molecules rather than RANKL, like ILs or PTHrP.

Whereas the optimal control framework is still used in numerous works, it prevails an urgent need to translate real quantitative data (economic cost of treatment, number of skeletal-related events and other side-effects of using the treatment, number of cells in a specific bone metastasis, and others) into the mathematical modeling language to convert these types of models into predictive tools towards the development of patient-personalized treatments. We visualize our results as a step forward to accomplish such goal.